I am a native English speaker, and I used to say this to my kids all the time....e.g if we were at a restaurant and they were playing with the ice in their drinks etc. "Leave the ice alone! You will get sticky fingers!"

They mean two different things. For example, there is a delivery of ice to the grocery store. You say Leave the ice here or there. You are in a restaurant and your children are taking the ice out of their sodas and playing with it making a sticky mess. You could say "leave the ice alone" or "leave the ice in your glasses"

A memory hook that may help some:
"Lasciate il ghiaccio" would mean "Leave the ice [behind]" implying that you "let it go" ... think of Anna asking Elsa to leave her ice palace.
"Lasciate stare il ghiaccio" implies "Do not touch the ice" like in Elsa telling Anna to not play with her ice palace.

That is just one of several ways to say this in English, varying by region. Leave/let [it] be/alone. In a suitable context, it could also be let [it] stay, but I can't imagine when this would refer to ice..